The Geomno!
by Leben ist Magie
Summary: Come, be regaled by a Scottish Columbus and a British Euclid while a young girl embarks on a bizarre quest for Geometry knowledge! This was my ninth grade Geometry project, but I don't have the heart to delete it. Complete.
1. Prologue

**Disclaimer: I don't own Dante's The Inferno, Gaardner's Sophie's World, or Alice and Wonderland, or Glencoe's Geometry Textbook. Satisfied? Wunderbar!  
This is my sad, crazed excuse of a Final Exam (Hey, my teacher said we could do ANYTHING!) If there are any learned Geometry or English Scholars out there, feel free to notify me of horribly rendered geometrical errors. **

**Prologue**

Stella crossed the threshold of her new Geometry classroom. The lights were turned off, and no-one was inside yet. _I must be terribly early_, she thought. Checking her watch, she concluded that it was not her, but her teacher that was tardy. She also wondered where her new classmates had gone to.

Sighing, she plopped down in one of the front desks, dropping her bag down beside her. Math was not her favorite subject, so having Geometry first period was, in Stella's position, a very undesirable way to start her new school year.

Five minutes later, Stella began to get suspicious. _The bell should be ringing soon, _she thought. _Where on earth were her teacher and classmates?_ Stella got up from her chair and looked around the room, hoping to find some clue as to her teacher's absence.

A wooden podium stood by the front board, and as Stella glanced at it, an object caught her eye. Lying on it was a leather bound book with gold writing, which proclaimed the book's title: The Geomno. Stella studied it curiously. The book certainly didn't look like a regular textbook, so it was very curious indeed that it would reside on top of the podium. She flipped the cover open to look at the cover page.

Suddenly, a great wind seemed to fill the room. The door slammed shut and the blinds covering the windows rattled deafeningly. Stella, very unnerved, glanced back at the book and her jaw dropped open as she saw the book's pages seemed to hollow out and form a darkened window right before her eyes. The torrent of wind grew stronger, seeming to tug on Stella's form, making it difficult for her to stay on her feet. Horror grew in Stella's mind as she realized that the window in the book was acting as a vacuum, trying to pull her into the pages!

Stella struggled as best she could against the strong gusts as it whipped about her, but she was fighting a loosing battle. The space between her and the book lessened, and finally she felt her feet being swept out from under her, and she plunged into an indeterminable darkness.


	2. Chapter One

Disclaimer: Same as before. 

**To all readers: ** _WOW_ you're still with me! I heart you corageous folk. Now on with the story!

**Chapter One**

Stella came to, feeling as though she had been run over by a train. She was sprawled out on her back, lying on some sort of soft surface. A sun shone down and warmed her face, while birds chirped overhead. Slowly, she opened her eyes, and at once was met by a bright robin's-egg-blue sky. A lush forest surrounded her, and the ground was covered by vivid, emerald green grass. A soft breeze ruffled the leaves on the trees, making them sound as if they were whispering amongst themselves in a language meant for only tree ears alone. The atmosphere was extremely peaceful, and for a few moments Stella tried to remember how she came by the place. A flashback of a roaring wind, a dark window, and a leather book went through her mind as her memory returned. Stella groaned and placed a hand on her head. Oh how it hurt! Oh how confused she was!

An outburst of jovial laughter broke through her muddled thoughts, and she sat up, startled. A man dressed in eighteenth century clothing glanced down at her from his position near a table laden with papers, pencils, and calculators. A whiteboard stood on his other side, scrawled with many different math symbols, which all appeared to be hastily drawn and very advanced math.

"I must admit, it is a very strange sight to see a girl appear out of air, thought I have seen stranger things." He spoke, with a deep, lightly accented voice. "Tell me how it is that you come by this remote location?"

"I-I'm not quite sure." Stella managed to say, still feeling quite shaken about her past ordeals. "I opened a book, and then a whirling wind filled toe room, and then I found myself here."

"Well, if you are here, that must mean that you are no longer where you started, correct?"

Stella nodded. "I am most definitely not in my Maths classroom."

"Well then! We can use inductive reasoning to find out what has occurred!" The man exclaimed, with a delighted smile.

Stella glanced at him, perplexed. "I beg your pardon? What is inductive reasoning? How will it help my predicament?"

"All will be explained! But first, I would like a formal introduction. My name is Goldbach. What is yours, young lady?"

"My name is Stella. Where exactly am I?"

"That can also be explained by logic!" The man gestured at the paper-strewn table. "And, I cannot be modest at all when I confess I am quite familiar with logic."

"How so?" Stella asked, intrigued by the strange man.

"Well, for many years now I have been working on a certain math problem. It is entitled Goldbach's conjecture. But let's save that for another time."

"Alright," Stella agreed.

"Now, I am going to teach you how to use inductive reasoning. Not only can you use this technique in finding out your location, but in many other everyday problems. Inductive reasoning is also very important in geometry."

"How can logic be important for math?"

"In order to prove that objects are really what they seem to be. It also has other uses, but we won't worry about those for now."

"Oh."

Goldbach gestured towards one of two stools near the whiteboard. "Would you like to take a seat?"

"Thank you," Stella said as she seated herself in one stool. Goldbach erased the scribblings on the whiteboard and produced a blue whiteboard marker from somewhere inside of his brown overcoat. Uncapping it, he turned to Stella.

"Part of the inductive reasoning is looking for patterns and making conjectures. A conjecture is an unproven statement that is based on observations. Do you have a conjecture for your present situation, Stella?"

"Well, I suppose I could say that I'm inside a book."

"Good. Now, a counterexample is an example that shows a conjecture is false. It is used to help prove conjectures."

"How do you prove that conjectures are true by proving them false?"

"Because you need to be able to prove a conjecture is true in _all _cases. If you find a counterexample, it means that it is not true in all cases."

"Oh, I see."

"Conjectures also don't have to be only right _or_ wrong. Some are not known to be either. They are called unproven. Goldbach's conjecture, the math problem I am working on, is an unproven conjecture,"

"How interesting!"

"I know of a method you are probably familiar with using, but not in math. Do you know what a conditional statement is?

"Oh of course! We use those in science all the time. It is a sentence in two parts, and contains a hypothesis and conclusion. A conditional statement is always written in if-then form, where the 'if' represents the hypothesis, and the 'then' represents the conclusion. (How does the book do it?)"

"Excellent. I couldn't have said it better myself. Anyhow, a conditional sentence can be used in math." Goldbach turned to the whiteboard and drew the letter P. "I am writing the letter P up here to represent the 'if' part of the statement." He turned and wrote the letter Q in a space near the letter P. "The letter Q will represent the 'then' part of the statement." He drew an arrow from P, pointing to Q. "The arrow symbolifies that P_ implies _Q. Now, if the arrow were to be reversed…" Gold berg erased the arrow and drew a new one, from Q to P "…then the statement would read Q implies P. This is also called the converse. Goldbach paused. "Are you getting all of this so far?"

"Yes." Stella nodded.

"Wonderful! Let's go on! Where was I…?"

"We were talking about converses."

"Ah yes. Now, the inverse is formed by negating the original statement. So, where it was P implies Q, it is now the negative of P implies the negative of Q." Goldbach erased the arrow and changed it back to what it was originally, so that P implied Q. "The symbol for negation looks like a little squiggly line, so when I draw one in front of P and one in front of Q, we get the inverse!" Goldbach drew two symbols, one in front of each letter. "Math can incorporate these statements easily. Let's say P represents 'If 2+n5', while Q represents 'then n would equal 3'. What would the three statements be?"

"Well, the conditional statement would be: If 2+n5, then n equal would equal 3. The converse of the statement would be: if n equals 3, then 2+n5. The inverse would be: if 2+n(does not)5, then n does not equal 3. Am I right?"

"Perfect!" Goldbach beamed. "You'll make a good mathematician some day, Stella. Now, moving on. A contrapositive statement combines the inverse and the converse together into one statement."

"Do you mean it reverses the hypothesis and the conclusion, as well as negating them?"

"Exactly! When we say it, the contrapositive would be negative Q implies negative P. Using the statement before, what would the contrapositive be?

"If n does not equal 3, then 2+n(does not)5."

"Correct! You should also remember that those statements don't necessarily have to be all true. In fact, it isn't expected for both an inverse and a contrapositive to be correct. However, the Inverse and the converse are expected to be incorrect, and the original statement and the contrapositive are expected to be true. When two statements are both true or both false, they are called equivalent statements. Now, we must drift away from conditional statements and learn about something new: biconditional statements!"

"What are those?"

"A biconditional statement is a phrase written as 'If and only if.'"

"How is that different from a conditional statement?"

"A biconditional statement states that something _must_ be true."

"I see. So if we used the same phrases from the earlier conditional statement and plug it in here, we would get: 2+n5 if and _only_ if n equals three, yes?"

"Correct!"

"But wouldn't we also say that it is similar to a converse?"

"Yes! That is what a biconditional statement is: equivalent to both a conditional statement and its converse."

"Oh, I see!"

"Well, you seem to understand inductive reasoning very well, so now I am going to introduce you to deductive reasoning."

"That term sounds familiar."

"Is should: you do it every day!"

"I do?"

"Yes! Deductive reasoning uses facts, definitions, and accepted properties in a logical order to write a logical statement."

"How does that differ from inductive reasoning?"

"Inductive reasoning uses previous examples and patterns to form conjectures."

"Ah! I understand."

"Wonderful! Now, there are two laws included in deductive reasoning."

"Laws? You mean like rules?"

"Yes! They are called the Law of Detachment and the Law of Syllogism."

"What do they say?"

"The Law of Detachment says that if 'P implies Q' is a true conditional statement, then both P and Q are true. The Law of Syllogism states that if 'P implies Q' and 'Q implies R' are--"

"Wait a second. What is R?"

"R is a third term. For example, we could say: if the sun is shining, then it is a beautiful day. That is P implies Q. Then we could go on to say: if it is a beautiful day, then we will have a picnic. That is Q implies R."

"Ah! I see now! Please, go on."

"All right. The Law of Syllogism states that if 'P implies Q' and 'Q implies R' are both true conditional statements, then 'P implies R' is also true."

"So it would be like saying if the sun is shining, then we will have a picnic?"

"Exactly! Very good, Stella. You did a wonderful job."

"Thank you. Does that mean you are done teaching me?"

"I'm afraid that's all I can teach you."

"But I still don't know where I am!"

"In Geomno, of course!"

"What is Geomno?"

"Geomno is the land in which you are in!"

"But how did I get here?"

Goldbach shrugged. "I don't know. It's not everyday a girl suddenly appears out of thin air, right under my nose! You might try asking someone else. I have some friends down the road." He pointed towards a well-worn dirt path that Stella had overlooked earlier. It led out of the clearing and deep into the woods. "You could try asking them."

"Oh, alright then! Thank you very much, Mr. Goldbach!"

"You're welcome, Stella. I hope you find what you're looking for."

"I hope so too! Goodbye!"

"Goodbye. May we meet again." Goldbach waved at her before turning back to his whiteboard. Stella turned around and set off down the road.


	3. Chapter Two

To you who have reviewed: THANK YOU THANK YOU THANK YOU!

**Chapter 2**

The woods were cool and dense. As Stella walked, she was shaded by the canopy of leaves overhead. A breeze played with her long, dark hair and yellow sundress. Goldbach had been very pleasant, and Stella hoped his friends would be just as nice. It was also good to have a friend in Geomno. By now, Stella didn't have a doubt that she was inside the book she had found on her teacher's podium. The question still remained of how she got there, and how she would travel back.

Presently, she came upon another clearing. To her surprise, she spotted a miniature version of an ancient Greek building she had once seen in her history book. What was it called again…? Ah yes, The Pantheon! What was a replica of it doing in the middle of a forest in a book?

Stella's curiosity overcame her and she ventured towards the stone structure. Did Goldbach's friends live here? He must be of a very odd sort, Stella thought, to live in such a house.

"γειά σας! (Hello there!)" A pleasant voice called out. Stella turned, to find an old, bearded man dressed in a toga.

"Who are you?" She asked, now thoroughly confused. Did the man speak English?

"Oh, you speak English! Pardon me, but Greek is my native language. My name is Euclid."

"Oh! I know who you are! We learned about you in history! You wrote The Elements!"

"Right, my girl! Now, what is your name?"

"I'm Stella."

"Well met, then. What brings you to my humble abode?" The man gestured at the miniature Greek building.

"I am trying to find out how I came to this place. Geomno, I mean. Do you know a man by the name of Goldbach?"

"Ah, yes of course! Jolly old chap…"

"He told me I might be able to find an answer with his friends."

"He did, did he?"

"Yes. Do you think he was referring to you?"

"Maybe, maybe. Tell me your story."

Stella explained the odd circumstances by which she came to Geomno. When she was done, Euclid wore a contemplative expression.

"Well, do you think you could help me?"

"I might be able to…but first! I like you, girl, because your quest for knowledge amuses me. So I will help you in this quest! Do you know what a definition is?"

"Yes of course! But what does this have to do with-"

"What is a definition, then?"

"It uses known facts to describe a new word."

"Precisely! And how about undefined terms, eh?"

"No. I really don't see what this has to do with helping me-"

"Patience, my dear girl, patience. Terms such as point, line and plane are classified as such. Do those words sound familiar to you?"

"Yes. A point is a small dot on a plane (think up a better definition!)"

"And what of the others?"

"A line extends in one dimension, and is represented by a straight line with two arrowheads. It extends without end in two directions. A plane extends in _two_ dimensions and is commonly represented by a shape which resembles a tabletop, although planes extend into infinity, also."

"Quite right. And how does a line differ from a line segment?"

"A line segment consists of two endpoints and a lone between."

"What about rays?"

"A ray has an initial point, and a line that extends into infinity in only one direction."

"Good. Can you define collinear points?"

"They are points that lie on the same line."

"And coplanar points?"

"Points that lie on the same plane."

"Bravo! It seems that you already know quite a bit, so I do not have much to teach you." Euclid paused for a moment, thinking. "Do you remember intersections?"

"Yes. Two or more geometric figures intersect when they share one or more points, and the point where they intersect is called the intersection."

"Very good. Now, I suppose Goldbach told you about reasoning and such?"

"Yes, he did."

"In the future, you will use those terms to help you prove statements. Another helpful tool you must learn is the usage of postulates and theorems."

"What are those?"

"Postulates, or axioms as they are sometimes called, are rules that are accepted without proof. Rules that are proved are called theorems."

"I see."

"You will be given those later on, but I wanted you to understand what they are first. Postulates and theorems also have names, such as the Pythagorean Theorem, which is very famous and, not to mention, useful."

"Ok."

"Now, getting back to what we were talking about before: points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is called the coordinate of a point."

"Oh yes! I learned this before!"

"Good! Well, I'm afraid that that's all I can teach you."

"Really? But…I still don't know the answer to my question!"

"What was your question? I apologize if you told me; my memory is failing in old age. I am 2,330 years old, you know."

"That's amazing! I didn't know people could live that long!"

"They can in Geomno!" Euclid winked. "Now what was your question?"

"How did I get into Geomno?"

"Ah! Well, you must have found some sort of enchanted book. I assume it serves as a portal between your world and this one."

"So Geomno is a different world?"

"Either that, or a different plane of your world."

"That's quite confusing."

"I suppose…have you ever heard of Quantum Mechanics?"

"No."

"Well, we'll save that for another time, then."

"Sure, but how do I get back home?"

"I don't know. I have a few friends down the road aways-maybe they can help you."

"Thank you, I will. Goodbye, Euclid!"

The Greek smiled. "Goodbye, Stella. I hope you are successful in your quest."

Stella smiled and waved as she headed off down the road again


	4. Chapter Three

**A/N: Disclaimer from before still stands for all chapters--me too lazy to rewrite over and over and over…**

**Please I would be ever so grateful if you would review! **

**Chapter 3**

As she hiked down the road, Stella's eyes took in the scenery around her. Slowly, the lush and dense forest gave way to a grassy plain. Long grass rustled in the cool breeze and the sky overhead was a clear, cloudless azure. Geomno is quite a lovely place, Stella thought. Her mind turned to all the events that she had experienced in the past few hours and marveled at how impossible everything seemed. Surprisingly, Stella hadn't felt the least bit anxious about finding a way home. She felt safe in Geomno, and had no doubt that all of her questions would be answered in time.

Stella's attention turned back to the scenery, as she noticed a house located a little ways off the road. It looked like a small Spanish mansion. Stella made her way over to it. "Hello?" She called out. This might be where Euclid's friend lived.

Stella heard the sound of a door opening somewhere above her head. "Buenos Dies, Senorita!" A cheerful voice called out. It was evident that Spanish was the person's native language.

Stella looked up and her eyes came across a middle-aged Spaniard with short black hair and glasses. The man wore a red jacket over a crisp white dress shirt. He was standing on a small balcony that protruded from the house, with his hands resting on the decorated iron fencing. Stella noticed that his hands were covered in ink stains, and figured he must be an artist or writer of some kind. "How may I help you?" the man asked with a warm tone.

"Hello there! Are you a friend of Euclid's?"

"Why yes, I am. My name is Santiago Calatrava. What is yours?"

"My name is Stella. It's nice to meet you Mr. Calatrava."

"You can just call me Santiago."

"Oh, thank you. I was wondering if you might be able to answer a question of mine."

"Well, I can certainly try."

"Ok! But in order for it to make sense, I think I need to tell you a little about the situation I'm in."

"That sounds reasonable. Go right ahead, Stella."

Stella proceeded to tell Santiago about how she came to Geomno and her experiences so far. When she was done, she asked her question. "Do you know how I can get back home?"

The Spaniard looked thoughtful for a moment. "I'm not quite sure. It might take me a while to formulate a theory."

"Oh." Stella couldn't hide the disappointment in her voice.

"But…while I do, would you like to come inside for a cup of chocolate?"

""Sure!" Stella smiled. She hadn't eaten or drank anything since breakfast, which now seemed to be eons in the past.

"Good. The door is over there." Santiago pointed to the left side of the house. "I'll meet you over there and let you in."

"Thank you very much." The man smiled in return before disappearing behind a curtain that draped in front of the door to the balcony.

Stella rounded the house and found Santiago waiting for her. She followed him as he led her into a warm, cozy kitchen. The Spaniard took a mug and ladled a helping of hot chocolate from a pot on the stove into it. "Here you are." He said as he handed the mug to Stella.

"Thank you." Stella took a sip. "Wow this is delicious!"

"You're welcome. That is my family's special recipe." Santiago winked. "Have a seat."

Stella pulled out a cushioned stool from the island in the middle of the kitchen and plopped down on it. Santiago switched off the stove and followed suit. "Well Stella," he said, "do you know of angles?"

"Oh, but of course!" Stella exclaimed. "An angle consists of two different rays that have the same initial point. The rays are the sides of the angle, and the initial point is the vertex of the angle."

"Muy bien!" Santiago beamed. "I am an architect, so angles are very important to my line of work. Now, angles that have the same measure are called congruent angles." Santiago whipped out a long piece of paper and a pen. "Let's say that I have an angle, A, and the measure of that angle is 90 degrees." On the paper, the Spaniard drew a right angle. "In writing, we would represent the measure of an angle by saying that the measure of angle A is equal to ninety degrees, or m A90 degrees."

"Ok." Stella took another sip of chocolate and set her mug down on the island, next to the paper.

"You also probably know that a 90 degree angle is called a right angle. It is represented by drawing a square at the angle, like this." Santiago drew a symbol on the angle.

(LEAVE SPACE FOR SYMBOL)

"Oh yea, I knew that already." Stella said.

"Then you also know that an acute angle is an angle whose measure is less than 90 degrees, and an obtuse angle is an angle whose measure is more than 90 degrees?"

"Of course. I also know that a straight angle is an angle whose measure is equal to 180 degrees, and is also known as a straight line."

"Precisely! Now, two angles who share a common vertex and a side, but have no common interior points are called adjacent angles. Interior points are points that are between each side of an angle. Exterior points don't lie in the interior of an angle."

"Alright." Stella nodded, digesting the information.

"Now I'm going to tell you about segment and angle bisectors."

"What are those?"

"Bisectors divide segments and angles into two congruent segments, or angles. The midpoint of a segment is the point that bisects the segment into two congruent segments. A segment bisector is a segment, ray, line, or plane that intersects a segment at its midpoint. Compasses and straightedges (the latter being a ruler without marks) are used to construct segment bisectors. An angle bisector is a ray that divides an angle into two adjacent angles that are congruent. Do you understand all that?"

"Yup!" Stella nodded.

"Muy Bien! Now, two angles whose sides form two pairs of opposite rays are called vertical angles." On the paper, Santiago drew an 'X' shape, numbering each angle.

(LEAVE SPACE FOR DRAWING)

"Angle one and angle three are both vertical angles."

"Doesn't that mean that angle two and angle four are also vertical angles?"

"Si! That's right. Now, linear pairs are two adjacent angles whose noncommon sides are opposite rays."

"Such as angle one and angle two? Do those two make a linear pair?"

"Yes. I see that you understand this perfectly, Stella! Congratulations!"

"Thank you."

"You're welcome. Are you familiar with complementary and supplementary angles?"

"Yes, I remember those. Two angles are complementary if the sum of their measure is 90 degrees. Two angles are supplementary if the sum of their measure is 180 degrees."

"Good! Now you must learn a few theorems and postulates, as well as their uses."

"I don't like memorizing things much."

"Memorization is a part of life, Stella! But, I do have a method to make the job a little simpler for you."

"Oh, thank you!"

"No worries. Now, let's begin…" Santiago drew a T-chart. He labeled the left column 'Statements', and the right column 'Reasons'. "This is the structure of a two-column proof. What we are going to do is actually _prove_ statements. To do this, we first need something to prove." Santiago drew three angles, which all looked to be similar. He labeled each with a letter: A, B, or C. "There are a few facts that you will be notified of before starting a proof. In our case, we know that angle A is congruent to angle B, and angle B is congruent to angle C. We want to know whether or not angle A is congruent to angle C."

"Can't we automatically assume from that information that the answer is yes?"

"Ah! But what tells you this? We need to figure it out with logic and rules, Stella. Let me introduce you to our first theorem: The Properties of Angle Congruence, or the PAC for short."

"What does it have to do with any of this?"

"The PAC states that angle congruence is reflexive, symmetric and transitive."

"What does that mean?"

"Let me explain this to you: The reflexive part of the statement means that for any angle A, angle A will always be congruent to itself."

"But isn't that obvious?"

"Yes, but it is an important thing to know when writing proofs. Think for a moment, Stella: all of the information in your head that leads you to believe that angle A is congruent to angle C is being written down in this proof. You get to see how your brain figures out why they are congruent, but out on paper."

"Oh, I see! I'm glad Goldbach taught me about logic now."

"Yes, logic is very useful in geometry. Now, the symmetric part of the PAC means that if angle A is congruent to angle B, then angle B is congruent to angle A. Do you understand?"

"Yes-it's kind of like a mirror."

"It is exactly that, Stella. The last part of the PAC, which is the transitive, states that if angle A is congruent to angle B, and angle B is congruent to angle C, then angle A is congruent to angle C."

"Well that's it, isn't it? We've just proved that angle A is congruent to angle C through the transitive of the PAC."

"Not quite. We need to write it down in a proof. The first step to writing a proof is to state the already obvious. Just watch for a moment." Stella watched as Santiago numbered down the left column, and then wrote in _angle A is congruent to angle B, angle B is congruent to angle C_ in the 'Statements' column. He then proceeded to number the right column, corresponding it to the left. Under the number one, he wrote in the word _Given_. "This is how a proof works. In the right column, you write statements. In the left column, you validate your reasons for the statement. The given will always be included in the proof."

"I see." Stella said, leaning over the paper. "What happens next?"

Santiago chuckled. "A great many things. Stating the obvious is always the first step. What is obvious in this problem, Stella?"

"Well…the measure of angle A is equal to the measure of angle B."

"Muy bien! And why is this?"

"Why-that is what a congruent angle is!"

"In a proof, it would be written as the definition of congruent angles." Santiago filled out the chart, and looked up. "What's next, Stella?"

Stella concentrated hard, her brow furrowing, looking for obvious things that could help her prove her point. "The measure of angle B is congruent to angle C?"

"What would be your reason? You must always have a reason."

"The definition of congruent angles."

"Si, Stella! And after that, what would you do?" He began filling out the third statement and reason for what she had supplied, listening for her answer to the next question.

"The measure of A is congruent to the measure of angle C? By…the transitive property?"

"The transitive property of equality. But, yes, you are right again!" He filled out the next line. "What is missing?"

"Well…oh! I see: angle A is congruent to angle C, by the definition of congruent angles."

"Wonderful! You have just completed your first proof, Stella. Congratulations!"

"Thank you!"

"I have but a few more things to introduce you to: perimeter, circumference, and area."

"Oh, but I know them already!"

"What are the formulas for perimeter and area for a square?"

"The perimeter equals four times the side length. The area is equal to the side length squared."

"How about the triangle?"

"The perimeter is equal to the sum of the side lengths, and the area is equal to one half of the base times the height."

"Muy bien, Stella! I am done teaching you now, and may I remark upon what a wonderful student you were?"

"Thank you!"

"I regret to say that I have not come up with much of an answer to your dilemma. The best thing I could say was to seek out the book that you traveled through."

"But the book is still in my Geometry classroom, which is in a different world! How can I reach it from here in Geomno?"

"Alas, but I do not know. However, can you prove that the book did not travel with you? In fact, can you prove of its location at all?"

"Well, no…"

"So you have not lost hope yet, my friend. I know of a person who lives near here, and they might be able to help you better than I could hope to. If you follow the road, you will find them."

"Oh thank you, Santiago! I am very grateful of your help. I hope I will see you again someday!"

"I have no doubt that our paths will cross again in the future. May you have luck in your search!"

"Goodbye!" Stella walked out the door and back on the path.


	5. Chapter Four

Chapter 4

The sun was now much lower than it had been before. Stella became slightly worried-suppose she would have to stay in Geomno overnight? Where would she find shelter? Stella had brought no money with her, and for all she knew, the currency in this land was something totally different from America.

Stella became distracted from her thoughts as she spied a very odd sight: there was the top of a ship's mast, complete with a crow's nest, peering over the crest of the next hill. How in the world did something that bizarre turn up in the middle of a plane? Stella hadn't seen any signs of water, except for a few small streams that wouldn't be able to allow a whole ship to pass through. She came over the crest of the hill and the sight she met with was very hard to believe. There was an entire wooden ship, set up in the middle of the plain. It's hull was buried half-way in the ground, and the sails were unfurled and snapped in the breeze. Upon closer inspection, Stella spied that the name of the ship was the Santa Maria. That sounded slightly familiar…

"Ahoy there!" A voice resounded from the ship. Stella looked up, to try to catch a glimpse of the caller. A man peered over the ship's railing. He had fuzzy blonde hair covered by an outrageously plumed hat, and a ruddy face. The rest of his clothing was just as extravagant, which looked rather foppish next to the background of a weathered ship.

"Hello!" She called, waving her hand. "Are you a friend of Santiago's?"

"Indeed I am, lass. Why do you ask?"

"I need to find a solution to a problem, and Santiago referred me to you."

"Ah! In need of assistance? I'll be glad to help! Here, come up on deck." The mad dropped a rope ladder over the side of the ship and Stella promptly climbed up. By the time she had reached the top and managed to drop somewhat ungracefully on the rough wooden deck. The man helped Stella to her feet.

"Thank you," said Stella.

"Ah! That is not a problem. So tell me, lass, what is your name?"

"I'm called Stella. Who are you?"

"My name is Christopher Columbus. I prefer it if you would call me Columbus." The man introduced himself somewhat brashly, with an air of superiority.

"Wow, I've read about you in my history book! It's very nice to be of acquaintance with you."

The man beamed, his ego sated. "Now, Stella, what is this problem of yours?" Columbus sat down on a small barrel, and gestured for Stella to do the same.

Stella told the man of her experiences from the time she left her geometry classroom to the present. "I am looking for a way home. Can you help me?" She pleaded.

Columbus mused for a moment. "Yes, I might be able to find some sort of safe passage for you…but first: why don't I teach you what I know? Oh, and it is getting quite near to suppertime. Would you like to join me?"

"Oh yes, thank you. That would be excellent." Stella could feel her belly rumble at the mention of food.

"Alright then! How does plum duff sound?"

"What is plum duff?"

"Ah! It is a traveler's dish worthy of the King's halls. You will be pleased, I assure you." Columbus led her to a closed hatch, and proceeded to open it and venture down into the darkness. Stella squinted in order to see, until he lighted a match. "You may come down, Stella. I keep a very clean ship!"

Stella nodded and followed after him, down through the hatch and through a narrow passageway. She came out to a room filled with cookware, an oven, stove, refrigerator, and a sink. Against one wall was a massive table with equally massive chairs. A delicious smell emanated from the oven. Columbus had been busy checking on whatever was baking when Stella came in. He peeled off his oven mitts (in the shape of lobsters) and drew out two of the large chairs. He sat down in one and gestured Stella to do the same. She sat down in the chair and Columbus lounged back, stretching his legs before him.

"What do you know about parallel and perpendicular lines?" He asked.

"Well, parallel lines are coplanar and don't intersect. Perpendicular lines intersect once and form right angles at their points of intersection."

"Skew lines are another form of lines. They do not intersect and aren't coplanar. A transversal is a line that intersects two or more coplanar lines at different points." Columbus strode over to a kitchen cabinet and took from it a piece of paper and a pencil. He came back to the table and plopped gracefully down into the chair, laying the paper out on the table. He grasped the pencil and drew three lines, labeling them and their angles:

"T is a transversal. The angles formed by the transversal have been appointed special names."

"What are they called?"

"Two angles (one and five) are corresponding angles, because they occupy corresponding positions. Alternate interior angles lie outside the two lines on opposite sides of the transversal, like angle one and angle eight. Angle three and angle six are alternate interior angles because they lie between the two lines on opposite sides of the transversal. Consecutive interior angles, like angle three and angle five, lie between the two lines on the same side of the transversal. Consecutive interior angles are sometimes known as same side interior angles.

"I see."

"Good. Now, these can become very important if you want to prove that lines l and m are parallel, or if you want to prove that, say, angle one and angle five are congruent. Before you learn how to prove them, however, you must learn a few postulates and theorems. The first is the _Corresponding Angles Postulate,_ or the CAP as I like to call it. The theorem states that if two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. The converse of that postulate is called the _Corresponding Angles Converse_. It says that if two lines are cut by a transversal so that corresponding angles are congruent, then the two lines are parallel."

"Ok."

"The next theorem is called the _Alternate Interior Angles Theorem_, or the AIA, which states that if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. The converse states that If two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel. Ready for the next one?"

"There are _more_?"

"I'll give them to you in a nutshell; they are all the same except for the differences in angles."

"Ok," said Stella, relieved.

"The _Consecutive Interior Angles Theorem_ (CIA) states that if two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. The _Alternate Exterior Angles Theorem_ (AEA) states that if two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. There are converses for both of these. Now we must cover the _Perpendicular Transversal_ (PT), which is slightly different from the others. It says that if a transversal is perpendicular to one of the two parallel lines, then it is perpendicular to the other. Do you understand those?"

"Yes. They are crystal clear!"

"Very good. Now, do you know about slopes?"

"Yes! It's rise over run. Is that important in geometry, too?"

"Yes. There are a few basics you must always remember about perpendicular and parallel lines in a coordinate plane. If two nonvertical lines are parallel, then they will most definitely have the same slope."

"Why only nonvertical lines?"

"Because all vertical lines are always parallel."

"Oh! I see! What about perpendicular lines?"

"In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slope is -1. Vertical and horizontal lines are always perpendicular."

"Ah-ha! I understand."

"That is very good! My lesson is now over. Would you like some plum duff?"

"Oh, yes, please."

"Very good." Columbus took the meal from out of the oven and cut two helpings with a kitchen knife. He scooped the food out on two plates. He also fetched silverware and two small tankards which he filled with cider. He handed a plate and tankard to Stella. "Here we are!" Columbus exclaimed happily, digging in to his meal with an enthusiastic gusto. Stella took an experimental bite and chews for a moment with a skeptical look on her face, which quickly turned into surprise. The food was delicious!

After they had finished the meal, Stella turned to Columbus. "Have you given any thought to my question?"

Columbus leaned back in his chair. "Yes. I think that you must find the book that you came through-The Geomno, you say it was called? Yes, you must find it. It will be somewhere in this world-of this I have no doubt. When you find the book, I think you will know what to do."

Stella could have jumped for joy. "Thank you very much, Columbus! I am indebted to you! Only one thing: where is the book?"

"That is a question I do not know. I have some friends over yonder, however. They might be able to help you along."

"Oh, ok then." Stella jumped up from her chair. "I am afraid I cannot delay, so thank you again for the meal-it was delicious! Goodbye, Columbus!"

"May lady luck be with you, Stella! Farewell!" Columbus waved with his plumed hat. Stella nodded one last time and made her way up through the hatch. She climbed down the rope ladder with much more ease than before, and made her way down the road. By now, the setting sun had stained the sky a blood red. Stella cast a worried glance over her shoulder. Who knew what things could lurk in the wilds at night in this strange land?


	6. Chapter Five

Chapter 5

Stella ventured on down the path. In the distance, she could see the beginnings of another wood. She sighed. Would it be safe to stay in the pains, or try to seek shelter in the woods? Stella didn't know how far away Columbus' friends might be, and if she hadn't come upon them by twilight then she would have to be prepared to weather the night alone.

Right then, Stella's eyes spied an odd figure off the path aways to her right. It appeared to be a roof, but when Stella squinted against the dim light she saw that it was, in fact, a…_triangle._ Or a triangular prism to be more exact. It was also huge-a model that was at least 30 feet in height. One more odd thing struck Stella: the triangle was _pink_. How much odder could Geomno get?

"Hello?" Stella called out.

"Why, hello there." A mild voice spoke out. Stella looked around, and her eyes settled on an old man with a long white beard. He looked very much like Euclid in the fact that he also wore a toga, but he was much older. His blue eyes sparkled with humor and wisdom. _He reminds me of Gandalf, from the Lord of the Rings_, Stella thought to herself.

"How may I help you, young lady?" The man asked kindly.

"My name is Stella. Are you a friend of Columbus'? I'm looking for a book…"

"It's nice to meet you Stella. My name is Pythagoras. What book are you looking for?"

"The title of it is The Geomno. Have you come across it?"

"I am afraid not. Why do you need it?"

"Without the book, I won't be able to find my way home."

"I see. Where exactly is home?"

Stella explained the details of her predicament. Pythagoras listened attentively to her story. When she was done, he mused. "I certainly know of a few places you could look for it. First, why don't we take a step back? I know of a few things I could teach you."

"Ok."

"Define a triangle."

"A triangle is a figure formed by three segments joining three noncollinear points."

"Very good. You can classify triangles by their sides and by their angles."

"How?"

"An equilateral triangle is a triangle which has three congruent sides. An isosceles triangle has at least two congruent sides. A scalene triangle had absolutely no congruent sides. If you wish to classify triangles by angles, there are acute triangles, which have three acute angles. Also, there are equiangular triangles, whose angles are all congruent, the right triangles, which have one right angle, and the obtuse triangles, which have one obtuse angle."

"I see."

"We must also learn the anatomy of a triangle. Their different parts all have names. Each of the three parts joining the sides of a triangle is called a vertex. The two sides sharing a common vertex are adjacent sides. Also, the sides of right triangles and isosceles triangles have special names. In right triangles, the sides that form the right angle are the lets of the right triangle. The side opposite the right angle is the hypotenuse of the triangle. In isosceles triangles, the sides that are congruent are called the legs. The third side is the base of the isosceles triangle."

"Ok." Stella said.

"Now, when the sides of a triangle are extended, other angles are formed. The three original angles are the interior angles. The angles that are adjacent to the interior angles are the exterior angles. Do you understand?"

"Yes, I do."

"Good! Now we must learn a few theorems and postulates."

"Does all of geometry revolve around those?"

"I'd say it was the other way around. Now, the _Third Angle Theorem_ states that if two angles of one triangle are congruent to two angles of a second triangle, then the third angles are also congruent. This is one method of proving that triangles are congruent. The _Side Side Side Congruence Postulate_ states that if three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. The _Side Angle Side Congruence Postulate_ states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. The _Angle Side Angle Congruence Postulate_ states that if two angles and a non-included side of one triangle are congruent to the two angles and the corresponding noncongruent side of a second triangle, then the two triangles are congruent. The _Hypotenuse-Leg Theorem_ states that if the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and leg of a second right triangle, then the two triangles are congruent. Do you understand all of that?"

"Yes."

"Good! Now I am going to test your knowledge. Come here." Pythagoras beckoned Stella over towards a table that was, incidentally, also pink, although it was shaped in a rectangle. Situated on top of it were a few sheets of clean, white paper and a pen. Pythagoras grasped the pen, and with a few quick strokes, had drawn six triangles on the paper.

"Look at the first pair of triangles, Stella. Are they congruent?"

"Yes."

"Prove it. Give me a reason; that is what I taught you the postulates and theorems for."

"Well, all three sides on the first triangle are congruent to the tree corresponding sides of the other triangle…so I'd have to say that the postulate to prove their congruence is the Side Side Side Congruence Postulate."

"Good! That is perfectly correct. What about the second pair?"

"They are congruent by the Angle Side Angle Congruence Postulate."

"And the third?"

"It can be proven congruent by the Hypotenuse-Leg Congruence Theorem."

"Very good! You have learned all that I have to teach you. Congratulations."

"Thank you. Can you answer my question now?"

"Yes and no. I am afraid I do not know the exact whereabouts of the book you are seeking. I do, however, have a friend who might. They live over yonder in the forest, and will be able to help you." Pythagoras glanced at the darkening sky and produced a lantern from under the table. "You had better take this. The forest can be very dark, even before the sun goes down." The lantern suddenly began to glow with a warm light. He handed it to Stella. "Good luck on your quest, Stella."

"Thank you very much! Goodbye, Pythagoras!" Stella waved as she backed up towards the road.

"Goodbye, Stella."

Stella turned her back on Pythagoras and towards the direction of the woods.


	7. Chapter Six

Chapter 6

As Stella entered the forest, she realized just how correct Pythagoras had proved to be. The forest was blacker than night. She became very grateful of the little lantern Pythagoras had given her. Surprisingly, it emanated no heat. She wondered if Geomno had more advanced inventions and contraptions than her home. Stella caught a glimpse of an odd looking figure down the road, and quickened her pace so that she might see it. The object turned out to be another huge pink triangle, but this one was not a prism. It was only 2-D and looked as if it were a road sign, but there weren't any turns or forks in the road. Stella looked more closely at it and observed that it had writing. She read the words out loud.

"_A segment bisector intersects a segment at its midpoint. A segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called a perpendicular bisector. A point is equidistant from two points if its distance from each point is the same. _

_Perpendicular bisector theorem:_" Stella groaned. Not another theorem! _"…If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. _

_The distance from a point to a line is defined as the length of the perpendicular segment from the point to the line. When a point is in the same distance from one line as it is from another line, then the point is equidistance from the two lines. _

_Angle Bisector Theorem: If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle." How interesting_, thought Stella

_"A perpendicular bisector of a triangle is a line that is perpendicular to a side of the triangle at the midpoint of the side. When three or more lines intersect in the same point, they are called concurrent lines. The point of intersection of the lines is called the point of concurrency. The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter of the triangle. _

_Concurrency of Perpendicular Bisectors of a Triangle: The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle._

_A median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. The three medians of a triangle are concurrent. The point of concurrency is called the centroid of the triangle. It is always inside the triangle._

_Concurrency of Medians of a Triangle: The medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side. _

_An altitude of a triangle is the perpendicular segment from a vertex to the opposite side. It can lie inside, on, or outside the triangle. Every triangle has three altitudes. The lines containing the altitudes are concurrent and intersect at a point called the orthocenter of the triangle. _

_Concurrency of Altitudes of a Triangle: The lines containing the altitudes of a triangle are concurrent. _

_A midsegment of a triangle is a segment that connects the midpoints of two sides of a triangle. _

_Midsegment Theorem: The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long" _Stella sighed. That was quite a bit to take in! She caught a few extra words at the base of the triangle which she hadn't bothered in reading before._ "I hope you find this helpful, Stella. Pythagoras." Of course_, Stella thought. Pythagoras did, after all, live in a triangle house. She took one last glance at the triangle before starting off again, musing over the facts she had just learned.

After many minutes of walking (perhaps hours), Stella became impatient. Where was Pythagoras' friend? Had she missed a turn? Stella become worried, until suddenly, right in front of her, she glimpsed a gate. Hurriedly, Stella crossed over to it.

The gate was constructed of thick oaken doors. It was very tall, and was wide enough to span the entire road. The odd thing about it, however, was the writing engraved in the doors. Stella read the intricately carved words out loud to herself.

"_All read, ye who wish to enter here…A polygon is a plane figure. It must be formed by three or more segments called sides, such that no two sides with a common endpoint are collinear. Each side must intersect exactly two other sides, one at each endpoint. Each endpoint of a side is a vertex of the polygon. A polygon is convex if no line that contains a side of the polygon contains a point in the interior of the polygon. A polygon that is not convex is called nonconvex, or concave. A polygon is equilateral if all of its sides are congruent. A polygon is equiangular if all of its interior angles are congruent. A polygon is regular if it is equiangular and equilateral. A diagonal of a polygon is a segment that joins two nonconsecutive verticals. _

_A parallelogram is a quadrilateral with both pairs of opposite sides parallel. A rhombus is a parallelogram with four congruent sides. A rectangle is a parallelogram with four right angles. A square is a parallelogram with four congruent sides and four right angles. A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are the bases. A trapezoid has two pairs of base angles. The nonparallel sides are the legs of the trapezoid. It the legs of the trapezoid are congruent, then the trapezoid is an isosceles trapezoid." _Stella finished reading.Experimentally, she pushed at the heavy wooden door. Stella was thoroughly surprised when it proved to open without so much as a squeak. Beyond the gate, the pathway led downhill. Stella could not see much beyond the warm glow of the lamplight, and was hesitant to follow the path. _Oh well_, she thought. _This road hasn't let me down yet; I've never encountered danger._ And with a shrug of her shoulders, she set off downwards.

As she rounded one particularly sharp corner, her eye caught on something. A flash of silver had shone in the darkness of the trees. Stella furtively hoped that it wasn't the eyes of some wild beast that owned that silver flash. In the distance, she began to detect a slight gurgling sound, such as one would find near a stream. Stella squinted into the darkness. It may have been her imagination, but Stella saw a speck of light in the distance, that didn't have the same warm glow of her trusty lantern.

As she drew nearer, Stella became almost positive that there was a light up ahead. She glanced at it every once in a while as she walked, trying to see what might be lurking in the light. Finally, she became near enough to see.

The source of light was a floating, glowing contraption that hung over a small pool. A small trickle of a waterfall fed the pool, and that was where Stella had detected the gurgling noise. A large wooden podium that Stella distinctly remembered from somewhere in her past sat near the lake, also bathed by the pool of light.

Stella gasped and ran towards the podium. Maybe_, if I could only hope…The Geomno might be there…_, she thought as she ran. Sure enough, the leather-bound book sat on top of the podium, looking exactly as Stella had remembered it from her first encounter. Slowly, she flipped open the cover.

The wind picked up at once, tugging at Stella's dress and hair. Moments later a black window grew from the book, and pulled Stella into its dark vortex…

The End


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